1,720,996 research outputs found
Non local branching Brownians with annihilation and free boundary problems
No full textWe study a system of branching Brownian motions on R with annihilation: At each branching time a new particle is created and the leftmost one is deleted. The case of strictly local creations (the new particle is put exactly at the same position of the branching particle) was studied in [10]. In [11] instead the position y of the new particle has a distribution p(x, y)dy, x the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [10] and non local branching as in [11] and prove convergence in the continuum limit (when the number N of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions. We use in the convergence a stronger topology than in [10] and [11] and have explicit bounds on the rate of convergence.Fil: De Masi, Anna. Universita degli Studi dell'Aquila; ItaliaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Presutti, Errico. Gran Sasso Science Institute; ItaliaFil: Soprano Loto, Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
A Note on Fick’s Law with Phase Transitions
No full textWe characterize the non equilibrium stationary states in two classes of systems where phase transitions are present. We prove that the interface in the limit is a plane which separates the two phases
Renewal properties of the d = 1 Ising model
Full text linkWe consider the d = 1 Ising model with Kac potentials at inverse temperature β > 1 where the mean field predicts a phase transition with two possible equilibrium magnetizations ± mβ, mβ > 0. We show that when the Kac scaling parameter γ is sufficiently small, typical spin configurations are described (via a coarse graining) by an infinite sequence of successive plus and minus intervals where the empirical magnetization is "close" to mβ, and respectively, - mβ. We prove that the corresponding marginal of the unique DLR measure is a renewal process
Highly Anisotropic Scaling Limits
No full textWe consider a highly anisotropic d=2 Ising spin model whose precise definition can be found at the beginning of Sect. 2. In this model the spins on a same horizontal line (layer) interact via a d=1 Kac potential while the vertical interaction is between nearest neighbors, both interactions being ferromagnetic. The temperature is set equal to 1 which is the mean field critical value, so that the mean field limit for the Kac potential alone does not have a spontaneous magnetization. We compute the phase diagram of the full system in the Lebowitz–Penrose limit showing that due to the vertical interaction it has a spontaneous magnetization. The result is not covered by the Lebowitz–Penrose theory because our Kac potential has support on regions of positive codimension
Symmetric simple exclusion process with free boundaries
Full text linkWe consider the one dimensional symmetric simple exclusion process with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle. At a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate, independently, the rightmost particle dies. We prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem.Fil: De Masi, Anna. Universita Degli Studi Dell Aquila; ItaliaFil: Ferrari, Pablo Augusto. Universidad de Buenos Aires; Argentina. Universidade de Sao Paulo; Brasil. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Presutti, Errico. Gran Sasso Science Institute; Itali
Truncated correlations in the stirring process with births and deaths
No full textWe consider the stirring process in the interval ΠN:= [-N,N] of Z with births and deaths taking place in the intervals I +:= (N - K,N], and respectively I -:= [-N, -N + K), 1 ⤠K < N. We prove bounds on the truncated moments uniform in N which yield strong factorization properties
Microscopic models for uphill diffusion
No full textWe study a system of particles which jump on the sites of the interval [1, L] of Z. The density at the boundaries is kept fixed to simulate the action of mass reservoirs. The evolution depends on two parameters lambda' >= 0 and lambda '' >= 0 which are the strength of an external potential and respectively of an attractive potential among the particles. When lambda' = lambda '' = 0 the system behaves diffusively and the density profile of the final stationary state is linear, Fick's law is satisfied. In this paper we show that when lambda' > 0 and lambda '' = 0 the system models the diffusion of carbon in the presence of silicon as in the Darken experiment: the final state of the system is in qualitative agreement with the experimental one and uphill diffusion is present at the weld. Finally if lambda' = 0 and lambda '' > 0 is suitably large, the system simulates a vapor-liquid phase transition and we have a surprising phenomenon, as studied in Colangeli et al (2016 Phys. Lett. A 380 1710-3) and Colangeli et al (2017 J. Stat. Phys. 167 1081-111). Namely when the densities in the reservoirs correspond respectively to metastable vapor and metastable liquid we find a final stationary current which goes uphill from the reservoir with smaller density (vapor) to that with larger density (liquid). Our results are mainly numerical, we have theoretical explanations yet we miss a complete mathematical proof
Scaling limit of a generalized contact process
No full textWe derive macroscopic equations for a generalized contact process that is
inspired by a neuronal integrate and fire model on the lattice .
The states at each lattice site can take values in . These can be
interpreted as neuronal membrane potential, with the state corresponding to
a firing threshold. In the terminology of the contact processes, which we shall
use in this paper, the state corresponds to the individual being infectious
(all other states are noninfectious). In order to reach the firing threshold,
or to become infectious, the site must progress sequentially from to .
The rate at which it climbs is determined by other neurons at state ,
coupled to it through a Kac-type potential, of range . The
hydrodynamic equations are obtained in the limit .
Extensions of the microscopic model to include excitatory and inhibitory neuron
types, as well as other biophysical mechanisms, are also considered.Comment: 25 pages, 0 figure
Free boundary problems in PDEs and particle systems
No full textIn this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDE's. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analysed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases. All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms. In general researchers interested in the relations between PDE’s and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics
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